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  1. 1. Cross-Tabs Continued Andrew Martin PS 372 University of Kentucky
  2. 2. Statistical Independence Statistical independence is a property of two variables in which the probability that an observation is in a particular category of one variable and a particular category of the other variable equals the simple or marginal probability of being in those categories. Contrary to other statistical measures discussed in class, statistical independence indicators test for a lack of a relationship between two variables.
  3. 3. Statistical Independence Let us assume two nominal variables, X and Y. The values for these variables are as follows: X: a, b, c, ... Y: r, s, t , ...
  4. 4. Statistical Independence P(X= a ) stands for the probability a randomly selected case has property or value a on variable X. P(Y=r) stands for the probability a randomly selected case has property or value r on variable Y P(X=a, Y=r) stands for the joint probability that a randomly selected observation has both property a and property r simultaneously.
  5. 5. Statistical Independence If X and Y are statistically independent: P(X= a , Y= r ) = [P(X= a )][P(Y= r )] for all a and r .
  6. 6. Statistical Independence
  7. 7. If gender and turnout are independent: Total obs in column m * Total obs in row v N = mv
  8. 8. Statistical Independence Total obs in column m * Total obs in row v N = mv 210 * 100 300 = 70 70 is the expected frequency. Because the observed and expected frequencies are the same, the variables are independent.
  9. 9. 150 * 150 300 = 75
  10. 10. Here, the relationship is not independent (or dependent) because 75 (expected frequency) is less than 100 (observed frequency).
  11. 11. Testing for Independence How do we test for independence for an entire cross-tabulation table? A statistic used to test the statistical significance of a relationship in a cross-tabulation table is a chi-square test (χ 2 ).
  12. 12. Chi-Square Statistic The chi-square statistic essentially compares an observed result—the table produced by the data—with a hypothetical table that would occur if, in the population, the variables were statistically independent.
  13. 13. How is the chi-square statistic calculated? The chi-square test is set up just like a hypothesis test. The observed chi-square value is compared to the critical value for a certain critical region. A statistic is calculated for each cell of the cross-tabulation and is similar to the independence statistic.
  14. 14. How is the chi-square statistic calculated? (Observed frequency – expected frequency) 2
  15. 15. Chi-Square Test <ul><ul><li>The null hypothesis is statistical independence between X and Y. </li></ul></ul><ul><ul><li>H 0 : X, Y Independent </li></ul></ul><ul><ul><li>The alternative hypothesis is X and Y are not independent. </li></ul></ul><ul><ul><li>H A : X, Y Dependent </li></ul></ul>
  16. 16. Chi-Square Test <ul><ul><li>The chi-square is a family of distributions, each of which depends on degrees of freedom. The degrees of freedom equals the number of rows minus one times the number of columns minus one. (r-1)(c-1) </li></ul></ul><ul><ul><li>Level of significance: The probability (α) of incorrectly rejecting a true null hypothesis. </li></ul></ul>
  17. 17. Chi-Square Test <ul><ul><li>Critical value: The chi-square test is always a one-tail test. Choose the critical value of chi-square from a tabulation to make the critical region (the region of rejection) equal to α. </li></ul></ul><ul><ul><li>(JRM: Appendix C, pg. 577) </li></ul></ul>
  18. 18. Chi-Square Test <ul><ul><li>The observed chi-2 is the sum of the squared differences between observed and expected frequencies divided by the expected frequency. </li></ul></ul><ul><ul><li>If χ 2 obs ≥ χ 2 crit. , reject null hypothesis. Otherwise, do not reject. </li></ul></ul>
  19. 20. Chi-Square Test <ul><ul><li>Let's assume we want to test the relationship at the .01 level. </li></ul></ul><ul><ul><li>The observed χ 2 is 62.21. </li></ul></ul><ul><ul><li>The degrees of freedom is (5-1)(2-1) = 4. </li></ul></ul><ul><ul><li>The critical χ 2 is 13.28. </li></ul></ul><ul><ul><li>Since 62.21 > 13.28, we can reject the null of an independent relationship. </li></ul></ul><ul><ul><li>Y (attitudes toward gun control) is dependent on X (gender). </li></ul></ul>
  20. 21. Chi-Square Test <ul><ul><li>The χ 2 statistic works for dependent variables that are ordinal or nominal measures, but another statistic is more appropriate for interval- and ratio-level data. </li></ul></ul>